High School

Find the derivative of \( y = 12x^4 + \frac{5}{x^7} - 8 \csc x \).

A. \( y' = 48x^3 - \frac{35}{x^8} + 8 \csc x \cot x \)

B. \( y' = 48x^3 - \frac{35}{x^8} - 8 \csc x \cot x \)

C. \( y' = 48x^3 - \frac{5}{7x^6} + 8 \csc x \cot x \)

D. \( y' = 48x^3 - \frac{35}{7x^6} + 8 \csc x \cot x \)

Answer :

Final answer:

The derivative of the function y=12x⁴+5/x⁷-8 csc x is 48x³ - 35/x⁸ + 8 cscx cot x, which corresponds to option 'a'. The process involves the application of the power rule for differentiation and understanding of the derivatives of trigonometric functions.

Explanation:

The subject of this question is calculus, specifically the process of differentiation. The function you're asked to find the derivative of is y=12x⁴+5/x⁷-8 csc x. Let's solve it step by step:

  1. For the first term 12x⁴, apply the power rule: bring the exponent in front and subtract one from the exponent. That gives us 48x³.
  2. For the second term 5/x⁷, rewrite it as 5x⁻⁷ and again apply the power rule: (-7)*5x⁻⁸ = -35x⁻⁸ or -35/x⁸.
  3. Finally, the derivative of -8 csc x is 8 csc x cot x. This is a standard derivative you might memorize, or derive each time using the quotient rule and the derivatives of sin x and cos x.

So, putting it all together, the derivative of the function y is equal to 48x³ - 35/x⁸ + 8 cscx cot x. Hence, option 'a' is the correct answer.

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